{"title":"带加法门的等深度量子电路的计算能力","authors":"Y. Takahashi, Y. Kawano, M. Kitagawa","doi":"10.1109/CEC.2003.1299569","DOIUrl":null,"url":null,"abstract":"We investigate a class QNC/sup 0/ (ADD) that is QNC/sup 0/ with gates for addition of two binary numbers, where QNC/sup 0/ is a class consisting of quantum operations computed by constant-depth quantum circuits. We show that QNC/sup 0/(ADD) = QNC/sup 0/(PAR), where QNC/sup 0/(PAR) is QNC/sup 0/ with Toffoli gates of arbitrary fan-in and gates for parity. Moreover, we show that QNC/sup 0/(ADD) = QAC/sup 0/(MUL) = QAC/sup 0/(DIV), where QAC/sup 0/(MUL) and QAC/sup 0/(DIV) are QNC/sup 0/ with Toffoli gates of arbitrary fan-in and gates for multiplication and division respectively. In the classical setting, similar relationships do not hold. These relationships suggest that QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD); that is, the use of gates for addition increases the computational power of constant-depth quantum circuits. To prove QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD), we present a characterization of this relationship by the one-wayness of a permutation that is constructed explicitly. We conjecture that the permutation is one-way, which implies QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD).","PeriodicalId":416243,"journal":{"name":"The 2003 Congress on Evolutionary Computation, 2003. CEC '03.","volume":"2 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2003-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On the computational power of constant-depth quantum circuits with gates for addition\",\"authors\":\"Y. Takahashi, Y. Kawano, M. Kitagawa\",\"doi\":\"10.1109/CEC.2003.1299569\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We investigate a class QNC/sup 0/ (ADD) that is QNC/sup 0/ with gates for addition of two binary numbers, where QNC/sup 0/ is a class consisting of quantum operations computed by constant-depth quantum circuits. We show that QNC/sup 0/(ADD) = QNC/sup 0/(PAR), where QNC/sup 0/(PAR) is QNC/sup 0/ with Toffoli gates of arbitrary fan-in and gates for parity. Moreover, we show that QNC/sup 0/(ADD) = QAC/sup 0/(MUL) = QAC/sup 0/(DIV), where QAC/sup 0/(MUL) and QAC/sup 0/(DIV) are QNC/sup 0/ with Toffoli gates of arbitrary fan-in and gates for multiplication and division respectively. In the classical setting, similar relationships do not hold. These relationships suggest that QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD); that is, the use of gates for addition increases the computational power of constant-depth quantum circuits. To prove QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD), we present a characterization of this relationship by the one-wayness of a permutation that is constructed explicitly. We conjecture that the permutation is one-way, which implies QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD).\",\"PeriodicalId\":416243,\"journal\":{\"name\":\"The 2003 Congress on Evolutionary Computation, 2003. CEC '03.\",\"volume\":\"2 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2003-12-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The 2003 Congress on Evolutionary Computation, 2003. CEC '03.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/CEC.2003.1299569\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The 2003 Congress on Evolutionary Computation, 2003. CEC '03.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/CEC.2003.1299569","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the computational power of constant-depth quantum circuits with gates for addition
We investigate a class QNC/sup 0/ (ADD) that is QNC/sup 0/ with gates for addition of two binary numbers, where QNC/sup 0/ is a class consisting of quantum operations computed by constant-depth quantum circuits. We show that QNC/sup 0/(ADD) = QNC/sup 0/(PAR), where QNC/sup 0/(PAR) is QNC/sup 0/ with Toffoli gates of arbitrary fan-in and gates for parity. Moreover, we show that QNC/sup 0/(ADD) = QAC/sup 0/(MUL) = QAC/sup 0/(DIV), where QAC/sup 0/(MUL) and QAC/sup 0/(DIV) are QNC/sup 0/ with Toffoli gates of arbitrary fan-in and gates for multiplication and division respectively. In the classical setting, similar relationships do not hold. These relationships suggest that QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD); that is, the use of gates for addition increases the computational power of constant-depth quantum circuits. To prove QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD), we present a characterization of this relationship by the one-wayness of a permutation that is constructed explicitly. We conjecture that the permutation is one-way, which implies QNC/sup 0/ /spl subne/ QNC/sup 0/(ADD).
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